Overcoming Shortcut Learning in a Target Domain by Generalizing Basic Visual Factors from a Source Domain

Abstract

Shortcut learning occurs when a deep neural network overly relies on spurious correlations in the training dataset in order to solve downstream tasks. Prior works have shown how this impairs the compositional generalization capability of deep learning models. To address this problem, we propose a novel approach to mitigate shortcut learning in uncontrolled target domains. Our approach extends the training set with an additional dataset (the source domain), which is specifically designed to facilitate learning independent representations of basic visual factors. We benchmark our idea on synthetic target domains where we explicitly control shortcut opportunities as well as real-world target domains. Furthermore, we analyze the effect of different specifications of the source domain and the network architecture on compositional generalization. Our main finding is that leveraging data from a source domain is an effective way to mitigate shortcut learning. By promoting independence across different factors of variation in the learned representations, networks can learn to consider only predictive factors and ignore potential shortcut factors during inference.

1 Introduction

Humans seamlessly categorize objects in the real world by their basic visual factors (e.g., shape, texture, color). For example, we perceive a red fire truck as a object with red color and the shape of a fire truck. We are able to do this because we have learned abstract concepts of shape and color, which easily generalize to common objects including unseen, out-of-distribution (OOD) data [28]. Unlike humans, modern deep neural networks (DNNs) do not possess a generalized notion of basic visual factors and therefore tend to perform poorly on OOD data. This is especially true when networks exploit shortcuts inherent in the data, i.e., when they excessively rely on visual factors that are easy to learn and predictive on in-distribution data but fail to generalize on OOD samples [8, 27]. For example, DNNs trained on a dataset in which all fire trucks are red might use color as a shortcut to recognize fire trucks while ignoring the more semantically meaningful attribute of shape. As a result, such a network may misclassify a yellow fire truck as a school bus.

In this work, we study the ability of DNNs to recognize OOD samples in the context of compositional generalization, i.e., the ability to understand unseen combinations of known elements. While a few approaches have already been proposed to improve generalization in compositional zero-shot recognition tasks [2, 22, 23], the performance of all these methods heavily depends on the number of combinations observed during training. This is due to the fact that a low number of seen combinations generates more shortcut opportunities [8]. A naïve remedy to shortcut learning is to collect missing combinations but this could be costly as some combinations can be rare in practice. In this work, we aim to investigate the underpinning of compositional generalization in DNNs and hence focus on the challenging and unexplored regime in which different visual attributes are fully-correlated in the training data (e.g. where one attribute such as color is a deterministic function of another one, such as shape). In the rest of this paper, we will refer to the domain in which we aim to achieve compositional generalization as the target domain.

Fig. 1.

Our task is to predict attribute/object labels in the target domain whose correlated training labels induce high shortcut opportunities. To improve generalization, we propose to augment the training set with a cheap source domain to learn representations of visual factors. In this given example, attribute and object in the target domain associate to color and shape factors respectively.

We propose a novel approach to mitigate shortcut learning in such a target domain by introducing a well-controlled source domain in the form of an additional training dataset, which can be cheaply generated. This source dataset is specifically designed to facilitate learning of basic visual factors by constraining them to be uncorrelated (all factor combinations appear uniformly during training). With this source dataset, we aim to learn independent representations of generic visual factors in order to improve OOD generalization in the target domain. Overview of our setup is shown in Fig. 1. We show practical benefits of such a simple and easy-to-generate source domain for improving the performance across multiple, more complex, target domains. Our approach provides a low-cost and effective strategy to improve compositional generalization.

Considering our source dataset, we employ DiagVib [6], a framework to generate datasets whose visual factors (i.e., shape, color, texture, lightness and background) can be customized (see Fig. 3). These factors are suitable for our purpose for the following reasons. Firstly, they are generic for common objects. Additionally, as suggested by previous studies [9, 27], some visual factors (e.g., background and texture) are likely to introduce spurious correlations while other factors (e.g., shape) are more robust when used for recognizing objects. By including these key factors in the source dataset, our models can learn to focus more on important factors, ignore potential shortcut factors and thus improve the models’ robustness to shortcuts.

Our contributions are as follows: firstly, we introduce a novel framework to improve compositional generalization in fully-correlated target domains. This is achieved by leveraging a source domain in order to alleviate shortcut learning. Secondly, we propose a simple network architecture exploiting the source domain to learn independent representations of visual factors. We also show that, if the target domain is not strictly fully-correlated, we can require less a priori knowledge for our approach. Lastly, we perform ablation studies to investigate effects of different source dataset configurations. Our main finding is that the source domain can act as a regularizer that encourages the internal representations of basic visual factors in DNNs to be less entangled. We show that this consistently improves compositional generalization across different target domains.

2 Related Works

Compositional Generalization (CG). We study CG in the context of compositional zero-shot learning, where the goal is to recognize images of unseen attribute-object combinations given only some combinations seen during training. A simple baseline VisProd [19, 22] uses multiple classifiers to predict attribute and object labels. More recently, [2, 14, 17, 21,22,23] learn to map images and labels (i.e., attribute-object combinations) to their joint feature space. It is commonly assumed that the same object can be seen in combination with certain number of attribute values during training. In our work, we study the corner case where the number of seen combinations is minimal, e.g., a fully-correlated attribute-object combinations setting, which introduces severe shortcut opportunities. A recent work [2] shows dramatic performance degradation of DNNs when reducing the number of seen combinations. The objective of this work is to mitigate this effect by incorporating a suitable source domain. We remark that, in most CG approaches, the label space has only two dimensions (i.e., attribute and object types). Instead, in our work the label space of the source domain is not restricted to two dimensions but can be as high-dimensional as the number of annotated factors. In particular, we consider basic visual factors as considered in [6], which are more likely to generalize across different tasks.

Domain Generalization. The aim of domain generalization is to learn models that generalize to unseen domains at test time. The problem we address in this work is therefore related to this line of research. More specifically, our problem setting loosely resembles Heterogeneous Domain Generalization (HeDG) [12, 13, 26, 30, 31], because we assume that the label distributions in the source and target domains have different, possibly disjoint, support (e.g., the labels represent objects in the source domain and animals in the target domain). Nevertheless, it should be noted that our work stems from a different motivation compared to the domain generalization literature. Unlike domain generalization, we do not assume the target data distribution to be unavailable during training. Instead, we assume to have access to a heavily biased sample from the target domain. From this perspective, our setup is also related to domain adaptation [5, 7], where training is performed on both the source and target domains together.

Learning Independent Representations. The topic of compositional generalization is also linked to the notion of disentanglement in generative modeling. The goal of disentangled representation learning is to construct a compact and interpretable latent representation, by discovering independent factors of variation (FoVs) in the data [4, 11, 24]. Most methods proposed in the disentanglement literature assume statistical independence between the factors of variation and perform learning without supervision. Thus, they are predominantly trained and evaluated on synthetic data where the ground truth FoVs are perfectly uncorrelated [16]. This is an idealized setting, which is almost never encountered in the real world. Therefore, the usefulness and generalization ability of these methods when the training data is biased remains unclear. For instance, a recent large-scale empirical study found that several state-of-the-art methods from the disentanglement literature fail to disentangle pairs of correlated factors [25]. Our work is also concerned with learning independent representations of basic visual factors, but, as opposed to prior works, we specifically focus on the problem of mitigating shortcut learning. It should be noted that, in contrast to unsupervised representation learning, we train representations of visual factors with factor annotations. However, we will discuss a scenario in which unsupervised representation learning can be integrated into our framework in Sect. 4.3.

3 Methodology

3.1 Problem Formulation

Our task measures generalization performance in the presence of shortcuts from the perspective of compositional generalization. Specifically, let us consider a target domain/dataset t, where each sample consists of an image $x_t \in \mathcal {X}_{t} \subset \mathcal {I}$ containing a single object. Such object is associated with one attribute: $y_t = (a, o) \in \mathcal {Y}_{t} = \mathcal {A}_{t} \times \mathcal {O}_{t}$, where $\mathcal {A}_{t} = \{a_1, a_2, \ldots \}$ and $\mathcal {O}_{t} = \{o_1, o_2, \ldots \}$ are sets of attribute and object type values respectively. In our task, not all attribute-object combinations are seen during training. The goal is to learn a model for the joint probability distribution $p(a, o\mid x_t)$ which yields good predictive performance on images of both seen and unseen attribute-object combinations.

The target dataset may be heavily biased, and thus naively training a classifier to predict object and attribute types can lead to shortcut learning resulting in poor OOD generalization. We define a dataset in a target domain, to be fully-correlated when each annotated attribute label appears together with only one annotated object label or vice versa. We aim to investigate whether introducing an additional, specifically designed, dataset from a different source domain s can discourage DNNs from exploiting shortcuts when trained for attribute and object prediction in the target domain. For this purpose, each image $x_s \in \mathcal {X}_{s} \subset \mathcal {I}$ in the source domain s is assumed to be labeled with a tuple $y_s = (f_1, f_2, \ldots , f_K) \in \mathcal {F}_s^{1} \times \mathcal {F}_s^{2} \times \ldots \times \mathcal {F}_s^{K}$, where K is the number of factors and each $\mathcal {F}_s^{k}$ denotes the set possible values for an individual basic visual factor capturing generic image properties such as shape, color, texture, etc.

To utilize source factor information in the target domain, we build an association matrix that models the connection between different source factor representations to target attribute or object. For fully-correlated target domains, we assume that the association is given as a priori knowledge regarding the possible source factor(s) the model is expected to rely to avoid shortcuts. For example, in the Color-Fruit dataset (Fig. 4c/4d), attribute and object can be manually associated to color and shape factors respectively. This a priori knowledge requirement can be relaxed in semi-correlated setting as presented in Sect. 3.4.

3.2 Network Architecture

Architecture. We study a simple baseline model (similar to VisProdNN [22]), which naturally lends itself to exploiting the availability of a source dataset. The baseline model (Fig. 2a) consists of an encoder backbone G and multiple prediction heads. It maps an input image to a global latent representation, which is then used to predict both the attribute and object labels. A crucial limitation of this model is that, without additional inductive biases, its latent representation will encode all predictive image factors. This makes the model more vulnerable to learning shortcuts, since it will be free to rely on easy to learn, predictive signals irrespective of their (in)ability to generalize to novel combinations [10].

Fig. 2.

Architectures with different image representations. a) Global: a single vector contains all visual clues. b) Factor: multiple factor representations encode different visual factors provided in source domain.

We extend the model of Fig. 2a by splitting the latent representation into multiple factor representations, as in Fig. 2b. As opposed to Fig. 2a, the encoder G produces K non-overlapping factor representations. Each representation is intended to contain only information related to its corresponding basic factor (e.g., shape, color, texture). The encoder’s output can then be written as $G(x) = Z = \begin{bmatrix} z_{1}&z_{2}&\ldots&z_{K} \end{bmatrix} \in \mathbb {R}^{D \times K}$ (D is the size of a factor representation).

The prediction heads are divided in two subsets for predicting labels in source and target domains respectively. While in principle we could feed Z as an input to all the prediction heads, as discussed above, this approach leads to poor compositional generalization. Instead, we introduce an additional inductive bias, namely that each source prediction logit should only depend on a single factor representation. Therefore, predictions for the source data $H_{s}$ are:

$$\begin{aligned} \hat{y}_{s} = H_{s}(Z) = \begin{bmatrix} h_{s}^{1}(z_{1})&h_{s}^{2}(z_{2})&\ldots&h_{s}^{K}(z_{K}) \end{bmatrix} \;, \end{aligned}$$

(1)

where $\hat{y}_{s} = \begin{bmatrix} \hat{y}_{s}^{1}&\hat{y}_{s}^{2}&\ldots&\hat{y}_{s}^{K} \end{bmatrix}$ contains the predicted factor values for a sample from the source domain. Ideally, each $\hat{y}_{s}^{k}$ should only depend on $z_{k}$ to discourage the latent representation from encoding information irrelevant to predict the k-th factor. Due to biases in the target domain, and in the absence of additional constraints, the architecture introduced above does not ensure invariance of every representation to the other factors. We explore different strategies to promote independence of the learned representations in Sect. 3.3.

In fully-correlated scenario, the association matrix can manually be defined as a binary matrix $A \in \left\{ 0, 1\right\} ^{K \times 2}$ where $A_{k1}$ and $A_{k2}$ are set to 1 only if the k-th factor informs the attribute and object prediction respectively. The representations of attribute ($z_a$) and object ($z_o$) can be obtained by $\begin{bmatrix} z_a&z_o \end{bmatrix} = ZA$. The prediction on target data can then be computed as follows:

$$\begin{aligned} \hat{y}_{t} = H_{t}(ZA) = \begin{bmatrix} h_{t}^{a}(z_a)&h_{t}^{o}(z_o) \end{bmatrix} \;, \end{aligned}$$

(2)

where $\hat{y}_{t} = \left( \hat{a}, \hat{o} \right)$ is a tuple of the predicted attribute and object labels.

Loss. To train the encoder and the predictors, we use a linear combination of the two loss terms $\mathcal {L}_{source} = \frac{1}{K} \sum _{\forall k} CE(\hat{y}_s^k, y_s^k)$ and $\mathcal {L}_{target} = \frac{1}{2}\sum _{l \in \left\{ a, o\right\} } CE(\hat{y}_t^l, y_t^l)$, where CE denotes the cross-entropy loss. $\lambda \ge 0$ is a hyperparameter weighting the importance of the regularizing loss term $\mathcal {L}_{source}$, which encourages a factor representation via the source samples.

Training. An equal number of samples from the source and target domains are sampled for every minibatch and fed to the network in order to compute $\mathcal {L}_{source}$ and $\mathcal {L}_{target}$ separately. The network is optimized via gradient-based minimization of the total loss. In this regard, all source samples will affect G and $H_s$ and all target samples will affect G and $H_t$.

3.3 Additional Constraints

The factor representations $\{z_k\}_{k=1}^K$ may still be correlated when using the loss and model architecture described above. Consequently, in the target domain, $z_a$ (attribute representation) may be predictive of the object label, and $z_o$ (object representation) may be predictive of the attribute label. These are unintended shortcuts that lead to poor compositional generalization. We explore two additional constraints to further encourage independence among factor representations: the Isolated Latent Constraint and the Cross Independence Constraint.

Isolated Latent (IL) Constraint completely prevents factor representations from being influenced by the target domain. While this suppresses the effect of biases in the target dataset, it may harm discriminative performance in the target domain. This constraint is implemented by stopping gradients from $\mathcal {L}_{target}$ to the encoder G.

Cross-Factor Independence (CI) Constraint promotes independence of factor representations, by adding a set of small auxiliary networks $\left\{ H'_{k_{1}k_{2}}\right\} _{k_{1} \ne k_{2}}$ for cross-factor predictions. While each $H'_{k_{1}k_{2}}$ is trained to predict $y_s^{k_{2}}$ from $z_{k_{1}}$, G is trained to produce Z such that all $H'$ are poor predictors. More details are presented in the Appendix A.8. Although the CI constraint only encourages independence with respect to the source domain, we investigate if this property can be transferred to target domains in Sect. 4.2.

3.4 Learning Factor Association Matrix A

In non-fully-correlated target domains (e.g., semi-correlated setting mentioned in Sect. 4.3), we can relax the requirement that the association matrix A must be manually given a priori and, instead, learn it in an end-to-end fashion together with the network parameters. To this end, we apply a continuous relaxation of the binary matrix A, and allow it to contain real numbers within [0, 1] such that each column sums to 1 (i.e. overall weightage across source factors) to maintain scales of the attribute/object representations using the softmax function.

We found that naïvely learning A without any additional constraints can lead to poor properties of the association matrix. Ideally, the association matrix should match the target attribute or object type to only one (or a few) robust source factor(s), and ignore factors vulnerable to shortcuts. For this reason, we propose to add an additional regularization $\mathcal {L}_{Reg} = \alpha \mathcal {L}_{Entropy} + \beta \mathcal {L}_{Suppress}$.

$\mathcal {L}_{Entropy}$ is the sum of the entropy values of both columns of A. Minimizing this entropy loss will reduce the number of source factors used to predict target properties, encouraging only robust factors to be considered when minimizing together with cross-entropy losses. On the other hand, $\mathcal {L}_{Suppress}$ applies a regularization along the rows of A to make sure no same source factor is predictive of both target predictions. In particular, for each row i, if its maximum value $A_{ij^{max}}$ is higher than a threshold $\tau$, all other entries will be suppressed by adding $\left( \sum _{j \ne j^{max}}{A_{ij}}\right) *\left( \text {sg}(A_{ij^{max}}) - \tau \right)$ to the loss term. The symbol $\text {sg}(A_{ij^{max}})$ indicates the stop gradient operation, such that minimizing $\mathcal {L}_{Suppress}$ only affects the cell whose values are not maximum in each row. Detailed experiments on the automatic learning of the association matrix are presented in Sect. 4.3.

4 Experiments

We conduct several experiments to understand how incorporating the source domain affects compositional generalization in scenarios which are vulnerable to shortcut learning. We start by describing our experiment setup below.

Fig. 3.

Illustration of DiagVib-Caltech (main source dataset) showing its multiple independent factors. More details of each factor is presented in the Appendix A.1. (Color figure online)

Datasets. We use the following datasets in our experiments:

DiagVib. [6] We extend the original framework to use shapes other than MNIST to increase variances. Our main configurations based on this framework are a) DiagVib-Caltech: With 50 non-animal shapes from Caltech101 [18] in 12 colors. 5 basic visual factors are available as shown in Fig. 3. This dataset is our main source dataset. We will later show that compositional generalization can be improved by this low-cost dataset even in more complex target domains. b) DiagVib-Animal: With 10 animal shapes from [3] in 10 colors on 3 backgrounds and scales (see Fig. 4a/4b).

Color-Fruit. This dataset is comprised of real fruit images (of 5 types) from the Fruit-360 dataset [20]. Additionally, we control colors of the fruits using the recolorization approach from [29] (see Fig. 4c/4d). More details on this dataset generation are described in the appendix A.10.

Fig. 4.

Samples from target domains. A column of each grid corresponds to an object type. Notice that each object (shape) corresponds to a single attribute (color) during training (see a, c and e) but not during testing (see b, d and f). (Color figure online)

AO-CLEVR. This dataset is proposed in [2] to benchmark compositional generalization. It contains 3 basic shapes in 7 different colors. We will use this dataset as an additional target domain. We simulate correlation between attributes and objects by limiting one color to appear with only one shape during training.

Color-Fashion. This dataset is originally proposed in [15] in which each image sample depicts a person dressed with cloth combinations. For each sample, cloth type and color segmentations are provided. In this paper, original images are cropped so that only one cloth type is appeared in individual images (see Fig. 4e/4f). 5 cloth types (T-shirt, skirt, jeans, shoes and dress) and 5 colors (Black, White, Yellow, Green and Blue) are selected from the original dataset.

For all target domains in our work, if a manual factor association matrix A is required for any algorithms, attribute and object types (i.e., shapes, fruit types, cloth types) are associated to color and shape factors respectively. This is intuitive since shape can robustly predict object types in general. Poorly assigned association can lead to poor performance as detailed in the Appendix A.4.

Evaluation. We evaluate compositional generalization in the open world setting as in [23] by computing predictive accuracies on seen and unseen attribute-object combinations as well as their harmonic mean (HM). One difference to [23] is that, for fair comparison, we do not use an additional hyperparameter (i.e. bias term) to calibrate the likelihood of predicting unseen attribute-object combinations (More details regarding the bias term are described in the Appendix A.6). During testing, any attribute-object combinations in the target domain can appear. All quantitative results are averaged across 6 different random training seeds.

Network Configurations. We follow the same convention as in [2, 22, 23] by using ImageNet pretrained features from ResNet-50 as network inputs. G, $H_s$ and $H_t$ are modeled as fully-connected networks. Factor-0 and FactorSRC refers to our architecture when a source domain is ignored and used, respectively. IL and CI constraints are appended as suffixes if the constraints are applied. Also, we benchmark the architecture with the global image representation (Fig. 2a). Similarly, Global-0 and GlobalSRC denote its configurations where a source domain is not incorporated and incorporated respectively. More implementation details can be found in the Appendix A.5. The code for our implementation is publicly available at https://github.com/boschresearch/sourcegen.

Baselines. We compare our approach against the following baselines (1) LabelEmbed+ [22]: a vanilla baseline, which performs recognition with a joint feature space for images and labels. (2) TMN [23], which employs automatic network rewiring conditioned on attribute-object pair hypotheses. (3) CGE [21], which exploits graph structure to regularize the joint feature space.

4.1 Compositional Generalization in Fully-Correlated Scenario

We investigate the impact of using an uncorrelated source domain (all factor combinations can appear uniformly) across different target domains whose labels are fully-correlated. We compare several variants of our approach against baselines that do not use a source domain. The results are shown in Table 1.

We begin by noting that, without a source domain (first five rows), baselines generally perform well only on the seen combinations but not on the unseen ones. E.g., Factor-0 on the Color-Fruit dataset has seen accuracy of $100\%$ compared to only $2.9\%$ on unseen combinations (Low seen accuracies of some baselines are discussed in the Appendix A.6). This occurs as these networks are trained only on the target datasets with correlated combinations so that they learn to excessively exploit the easiest predictive visual factors present in the datasets. On the unseen combinations, these predictive factors do not necessarily generalize to the intended labels. This degrades the generalization of the networks.

In contrast, using a source domain (see FactorSRC variations) consistently improves the HM accuracy by increasing accuracy on unseen combinations, at the expense of a partial loss of performance on seen combinations. For example, the FactorSRC-IL baseline on the Color-Fruit dataset has a seen accuracy of $95.5\%$, which is lower compared to baselines that do not use the source domain, but in turn exhibits the highest unseen accuracy ($40.7\%$). This shows a reduction of shortcut learning. The same trend holds for all other datasets we consider, with different seen/unseen accuracy trade-offs across datasets.

Compared to previous works, our results show that a single and simple source domain improves generalization performance on unseen combinations across different target domains. An alternative naïve solution would be to collect data corresponding to unseen combinations directly in the target domain and include them in the training set. However, this is in practice not a viable solution: not only is collecting data in a real-world target domain expensive but, perhaps more importantly, the biases that affect the training set are often unknown, which makes collecting an uncorrelated dataset difficult in practice. Rather, generalization improvement on those target domains can be achieved without much additional costs if we have a universal (a source domain can be used for multiple target domains) source domain at hand that can be generated cheaply. One open problem is the trade-off between generalization and in-distribution accuracy, which in some cases is still sub-optimal, especially when the target domain has a large domain shift from the source domain (see the drop of seen accuracy of the FactorSRC-IL on the Color-Fashion dataset). Improving this trade-off is an open research question leaving a scope for improvement in future works.

Global vs Factor Image Representations. Our results suggest that a factor representation is essential to exploit the source domain for compositional generalization. In fact, all GlobalSRC variations, in spite of incorporating the source domain during training, exhibit significantly lower unseen accuracy compared to the FactorSRC variants. This is due to the fact that the global representation contains information about all visual factors that are relevant for the object-attribute prediction task, thus offering easy-to-learn shortcuts that harm generalization performance. In contrast, by using factor representations, we promote factor disentanglement such that shortcuts are harder to learn.

Table 1. Accuracies on DiagVib-Animal, Color-Fruit, AO-CLEVR (each has random chance accuracy of 1%, 4% and 4.7% respectively) and Color-Fashion target domains. DiagVib-Caltech is used as the source domain.

Table 2. Cross prediction accuracies on DiagVib-Animal. These are obtained by using each associated factor representation ($z_a$ or $z_o$) to predict each target label (attribute or object) with a linear model. Ideal independence representations will have high direct prediction accuracies (predict their own labels well) but low cross prediction accuracies (predict others’ labels poorly).

4.2 Impact of Additional Constraints

We investigate the role of IL and CI constraints. We begin by noting that FactorSRC-IL gives the best HM accuracies across all target domains. Our hypothesis is that this result can be explained by a larger cross-factor information flow into the learned factor representations when adding the CI constraint compared to IL. To quantitatively measure the magnitude of cross-factor leakage, we extract $z_k$ from all test samples and use them to predict all labels ($y_s^1, y_s^2, \ldots y_s^K$ for a source domain or $\hat{a}, \hat{o}$ for a target domain) with linear models. Ideally, $z_k$ should predict its associated label (direct prediction) well but should fail to predict other labels (cross-prediction). We present these results in Table 2.

First, we are interested whether the CI constraint can encourage independence, not only in the source domain, but more importantly in the target domain. We observe that, while independence is promoted by FactorSRC-CI in the source domain well (see the Appendix A.3), in the target domain, cross-prediction accuracies decrease only slightly compared to FactorSRC. Thereby, we can infer that the independence enforced by $\mathcal {L}_{H'}$ in the source domain is not necessarily transferable to the target domain. One possible reason is that factor representations are still affected by the dataset biases in the target domain via $\mathcal {L}_{target}$.

On the other hand, dataset bias in the target domain cannot affect the factor representations with the IL constraint. In Table 2, although an explicit independent constraint is not introduced, the factor representations are less entangled in the target domain, which is indicated by lower cross-prediction accuracies (while high direct-prediction accuracies are preserved). The independence which is indirectly encouraged only by $\mathcal {L}_{source}$ enables shortcut-robust factor representations resulting in better HM accuracies in Table 1.

4.3 Learning Association Matrix for Semi-correlated Scenario

The association matrix A must be known a priori for fully-correlated target domains because the target data alone does not contain enough information to distinguish object types from their attributes. For a semi-correlated target domain, a more general solution is viable, in which the association matrix is learned. In this case, each object type is observed in combination with at least two attribute values. The additional combinations make it possible to distinguish attribute from object type. In order to learn the association matrix A, we adopt the algorithm introduced in Sect. 3.4.

Table 3. Accuracies on various target domains in semi-correlated scenarios.

Table 3 reports performance of different algorithms. FactorSRC-IL-LA and FactorSRC-IL-LA^R indicate algorithms that learn the association matrix without and with association regularization respectively. Results suggest that naïvely backpropagating through the matrix A is not an effective strategy as indicated by the large gap between HM accuracy of FactorSRC-IL and FactorSRC-IL-LA. This is due to an undesired property of the association matrix which we will later investigate. Fortunately, this undesired property can be alleviated by incorporating our proposed regularization constraints during training, as indicate by the comparable performance of FactorSRC-IL and FactorSRC-IL-LA^R.

Fig. 5.

Factor association matrices learned from different target datasets with various approaches resulted from representative runs. Manual Association Matrix in Figure a is only used for our manual association (not the groundtruth).

To qualitatively assess the correctness of learned associations, we visualize A as heatmaps in Fig. 5a. In addition, we compare the association matrix that we manually assigned in FactorSRC* (Fig. 5a) and the learned ones. First we consider the simplest target dataset, DiagVib-Animal. In this case, FactorSRC-IL-LA, which does not apply any regularization, fails to retrieve the association matrix (Fig. 5b). Even if the association matrix has the high weights that associate shape factor to shape target and color factor to color target correctly, the matrix is still far from sparse, making the model vulnerable to shortcuts (see results in Table 3). On the other hand, when regularization constraints are introduced in FactorSRC-IL-LA^R, the learned matrix is more sparse and closer to the manually assigned matrix (see Fig. 5c), which results in lower shortcut vulnerability of the model and higher compositional generalization performance.

In the more realistic case of the Color-Fruit dataset, we find that with multiple random seeds, FactorSRC-IL-LA^R can converge to two possible configurations for the estimated matrix A. The first configuration (Fig. 5d) is close to our manual association as it matches the fruit type to the shape factor. Another configuration, on the other hand, associates the fruit type to the texture factor (Fig. 5e). This is not surprising, since in the color-fruit dataset, both shape and texture factors are predictive of fruit type. The same observation has already been made in the context of conventional image classification [9] especially with complex object shapes. We observed the same behaviour in the case of the Color-Fashion dataset where the learned matrix associates garment type to the texture factor (Fig. 5f). Another advantage of our approach indicated by this observation is that it yields higher model interpretability, as we can understand which visual factors are important for network predictions.

The fact that learning of the association is possible when target datasets are not fully-correlated makes our approach well suited to practical applications. As the association is determined automatically, minimal a priori knowledge is required. In other words, factor representations can also be learned from images by any approaches including unsupervised representation learning that disentangle image representations into multiple independent factor representations [11].

4.4 Properties of the Source Domains

We also investigate the properties of source domains which encourage generalization. Our main findings can be summarized as follows: first, basic visual factors represented in the source domain should be sufficiently diverse and aligned with visual properties of the target data. Second, the fact that all factor combinations are available in the source domain during training is crucial. This allows deep networks to learn meaningful representation for each factor. Lastly, large intra-class variation of factors is also important to encourage better generalization. More details of our ablation studies can be found in the Appendix A.2.

5 Conclusion

We study vulnerability of DNNs to shortcuts by evaluating their compositional generalization on target domains with correlated attribute-object combinations. We provide empirical evidence that incorporating an additional source domain can improve generalization on unseen combinations on target domains. The source domain enables certain networks to represent inputs in terms of multiple independent visual factors. From our findings, the impact of the source domain on compositional generalization relies on two major conditions: (1) Choice of network model: networks should have internal representations in which visual factors are disentangled and independent with respect to target domains (2) Choice of source domain: the source domain should be uncorrelated and cover main basic factors. The fact that our source domain is simple also shows a practical benefit that performance on certain target tasks can efficiently improve using an easy-to-generate source domain. This is relatively cheaper compared to acquiring samples from complex target domains. If target domains are not fully-correlated, some requirements of manual labels/annotations can be relaxed, leading to more practical applications of this work. We hope this work will serve as an inspiration to integrate inductive biases in the forms of datasets and network design.